１．北京行きの航空便は一日に8便（が）あります。The original sentense didn't have が, but I suppose it should be there.

No, you do not put が or を after a counter. 8便あります is correct.

２．山本は今出かけております。What does おりますmean here? What's its usage?

It's the humble form of います. In this case, the speaker is talking about someone else in his company to someone outside of the company, so he uses humble forms (talking about the in-group to the out-group).

３．日本の山では富士山が有名です。この（その）山には毎年３０万人が登ります。In this sentense , この is used, but can I use その as well?

No. その, when talking about a topic like this (rather than a concrete thing you are pointing at) means something that the person you are talking to has mentioned. この is used for things you have introduced, and あの is used for things you both know about.

４．申し込むには一定の資格が要る。Why there is a verb before に？I wonder whether 申し込み is better?

verb+には means "for the purpose of doing [verb]".

５．あくis instransitive verb, あけるis its transitive verb and あける is also its できform. And all できforms are instranstive verb, so is it contradictory here?

(the "でき form" is usually called the "potential form").

あく's potential form is not used (in general). Generally, the only verbs that have potential forms are ones that you have control over. Something like 窓が開く is not something you have control over, so 窓が開く can mean "The window can open" or "The window does open" or "The window will open". (Typically these verbs you have control over will take an object with を.)

あける is technically ambiguous, because it is a transitive verb "to open something", but also it is theoretically the (intransitive) potential of あく. As I said, though, verbs like あく are rarely used in the potential.

１．北京行きの航空便は一日に8便（が）あります。 The original sentense didn't have が, but I suppose it should be there.

No, you do not put が or を after a counter. 8便あります is correct.

Well, が can be used or omitted here actually. I personally feel が appears basically and can be omitted in this case... I don't know the reason but I feel ease to see a particle between a noun and a verb...

I think が appears or is omitted often in a subclause. And probably sometimes it should not be omitted (except for casual conversation).Ex. 一日に8便(が)あるのは北京行きの航空便だけです。 (can be omitted)Ex. 明日の公演は私が話します。(cannot be omitted)

I'm introducing these just from my feeling, not from logic. I'm sorry if my examples are not appropriate here...

あける is technically ambiguous, because it is a transitive verb "to open something", but also it is theoretically the (intransitive) potential of あく. As I said, though, verbs like あく are rarely used in the potential.

Interesting. Yeah, 開けられる is used for the potential, probably in order to avoid the ambiguity.

６．Which is 思う's できform? 思われる or 思える?

It would be 思える but I can't recall ever seeing this before.

It's theoretically possible to use 思われる for the potential but it is used mostly for the passive.(<人>には)～は/が～と思われる。(perhaps it should be passive...)But interestingly (and strangely), 思える is actually used just the same as passive. The 思われる and the 思える look to be interchangeable. (<人>には)～は/が～と思える。

These expressions are rather stiff. It sounds like in an official report or academic papers, to avoid using the sentence's subject, I or we (maybe it's because it looks objectively and not a fiction).

思われる is often used as a honorific as well. <人>は～と思われました。思うことができる may be the clearest way to express the potential.

A passive form of 思う is technically 思われる.A potential form of 思う is technically 思われる and 思える.

Actual use of a passive form of 思う is 思われる. (And seemingly 思える as well, but I may be wrong.)Actual use of a potential form of 思う is 思える(with a little hesitation) and 思うことができる(clear).

But sorry, probably I said too much. 思える and 思われる are not that stiff. They sometimes appears in any decent documents or conversation. I think it's not wasting time to remember these words. Actually, I think 思うことができる is far less frequent to see or hear.

As to あけられる, which is its potential form? あくor あける？How about passive form of あける? Also あけられる?

The potential form of あける is あけられる (in recent colloquialism, あけれる is used sometimes)I can open the lid.

The passive form of あける is あけられる as well.The lid can be opened.

The potential form of あく is technically あけられる (in recent colloquialism, あけれる is used sometimes)But I believe the potential form of あく (The lid can open), ie あけられる is hard to distinguish with the potential form of あける ie あけられる (The lid can be opened) because they look the very same.

The passive form of あく is technically impossible, maybe. (The lid opens. -> no passive)あく is an intransitive verb and probably the passive form is not applicable.

...But in that situation, I think what you want to say is actually the passive sentence with a transitive verb あける. In this case, the passive form of あける is あけられる (the lid can be opened).

Please note that あく is not a transitive verb. わたしはふたをあく and わたしはまどをあく are not correct. わたしはふたをあける and わたしはまどをあける are correct and ふたがあく and まどがあく are correct.